Courant Institute of Mathematical Sciences

About: Courant Institute of Mathematical Sciences is a education organization based out in New York, New York, United States. It is known for research contribution in the topics: Nonlinear system & Boundary value problem. The organization has 2414 authors who have published 7759 publications receiving 439773 citations. The organization is also known as: CIMS & New York University Department of Mathematics.

SE-Sync: a certifiably correct algorithm for synchronization over the special Euclidean group

Abstract: Many geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown poses \(x_{1},\ldots ,x_n \in \text {SE}(d)\) given noisy measurements of a subset of their pairwise relative transforms \(x_{i}^{-1}x_{j}\). Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics) and camera motion estimation (in computer vision), among others. This problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to effciently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides the exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction effciently. We combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so more than an order of magnitude faster than the Gauss-Newton-based approach that forms the basis of current state-of-the-art techniques.

A One-Dimensional Model for Dispersive Wave Turbulence

Abstract: A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number.

In silico methods for drug repurposing and pharmacology

Abstract: Data in the biological, chemical, and clinical domains are accumulating at ever-increasing rates and have the potential to accelerate and inform drug development in new ways. Challenges and opportunities now lie in developing analytic tools to transform these often complex and heterogeneous data into testable hypotheses and actionable insights. This is the aim of computational pharmacology, which uses in silico techniques to better understand and predict how drugs affect biological systems, which can in turn improve clinical use, avoid unwanted side effects, and guide selection and development of better treatments. One exciting application of computational pharmacology is drug repurposing-finding new uses for existing drugs. Already yielding many promising candidates, this strategy has the potential to improve the efficiency of the drug development process and reach patient populations with previously unmet needs such as those with rare diseases. While current techniques in computational pharmacology and drug repurposing often focus on just a single data modality such as gene expression or drug-target interactions, we argue that methods such as matrix factorization that can integrate data within and across diverse data types have the potential to improve predictive performance and provide a fuller picture of a drug's pharmacological action. WIREs Syst Biol Med 2016, 8:186-210. doi: 10.1002/wsbm.1337 For further resources related to this article, please visit the WIREs website.

The Integument of Water-walking Arthropods: Form and Function

Abstract: We develop a coherent view of the form and function of the integument of water-walking insects and spiders by reviewing biological work on the subject in light of recent advances in surface science. Particular attention is given to understanding the complex nature of the interaction between water-walking arthropods and the air–water surface. We begin with a discussion of the fundamental principles of surface tension and the wetting of a solid by a fluid. These basic concepts are applied to rationalize the form of various body parts of water-walking arthropods according to their function. Particular attention is given to the influence of surface roughness on water-repellency, a critical feature of water-walkers that enables them to avoid entrapment at the interface, survive the impact of raindrops and breathe if submerged. The dynamic roles of specific surface features in thrust generation, drag reduction and anchoring on the free surface are considered. New imaging techniques that promise important insights into this class of problems are discussed. Finally, we highlight the interplay between the biology, physics and engineering communities responsible for the rapid recent advances in the biomimetic design of smart, water-repellent surfaces.

Mean-Field Critical Behaviour for Percolation in High Dimensions

Abstract: The triangle condition for percolation states that\(\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} \) is finite at the critical point, where τ(x, y) is the probability that the sitesx andy are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thed-dimensional hypercubic lattice, ifd is sufficiently large, and (ii) in more than six dimensions for a class of “spread-out” models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values\((\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2)\) and that the percolation density is continuous at the critical point. We also prove thatv2 in (i) and (ii), wherev2 is the critical exponent for the correlation length.